Consider an Hamiltonian function $$ H(q_1,q_2,p_1,p_2)=p_1\, F_1(q_1,q_2) + p_2\, F_2(q_1,q_2). $$ Assume $q_2=g(q_1)$ for some function $g$.
I am interested in seeing what does this property imply on the dual variables $p_1,p_2$.
First, we have: $$ \dot p_1(t) = \frac{\partial H}{\partial q_1}(q_1(t),q_2(t),p_1(t),p_2(t)) = \frac{\partial H}{\partial q_1}(q_1(t),g(q_1(t)),p_1(t),p_2(t)) $$
Second, define $\tilde H$ such that: $H(q_1,q_2,p_1,p_2) = \tilde H(q_1,p_1,p_2)$ where $q_2$ has been substitute by $g(q_1)$. Then $$ \begin{array}{l} \frac{\partial \tilde H}{\partial q_1}(q_1,p_1,p_2) &=\frac{\partial H}{\partial q_1}(q_1,g(q_1),p_1,p_2) + g'(q_1)\, \frac{\partial H}{\partial q_2}(q_1,g(q_1),p_1,p_2) \\ &= \dot p_1 + \mathbf{g'(q_1)\, \frac{\partial H}{\partial q_2}(q_1,g(q_1),p_1,p_2)} \end{array} $$ but what does correspond this "correcting" term in bold ?
What can we say more ? Is it possible to define an Hamiltonian dynamics with only one state variable ?